Constructing minimally 3-connected graphs

TL;DR

This paper introduces an algorithm for constructing all non-isomorphic minimally 3-connected graphs using vertex splitting and edge addition, based on cycle analysis and isomorphism elimination techniques.

Contribution

It presents a novel algorithm that systematically generates minimally 3-connected graphs by extending smaller graphs while ensuring non-isomorphism.

Findings

Successfully constructs all non-isomorphic minimally 3-connected graphs for given sizes.

Uses cycle-based methods to verify 3-compatibility after graph operations.

Employs isomorphism certificates to eliminate duplicates.

Abstract

A $3$-connected graph is minimally 3-connected if removal of any edge destroys 3-connectivity. We present an algorithm for constructing minimally 3-connected graphs based on the results in (Dawes, JCTB 40, 159-168, 1986) using two operations: adding an edge between non-adjacent vertices and splitting a vertex. In order to test sets of vertices and edges for 3-compatibility, which depends on the cycles of the graph, we develop a method for obtaining the cycles of $G'$ from the cycles of $G$, where $G'$ is obtained from $G$ by one of the two operations above. We eliminate isomorphs using certificates generated by McKay's isomorphism checker nauty. The algorithm consecutively constructs the non-isomorphic minimally 3-connected graphs with $n$ vertices and $m$ edges from the non-isomorphic minimally 3-connected graphs with $n-1$ vertices and $m-2$ edges, $n-1$ vertices and $m-3$ edges, and $n-2$ vertices and $m-3$ edges.